Next, we split off vertex \(s\)  using an iterative algorithm that again uses the extreme sets tree. At a high level, this splitting off algorithm follows a similar structure to the Benczúr-Karger algorithm, but with a couple of crucial differences that improves the running time from \(\tilde{O}(n^2)\)  to \(\tilde{O}(m)\) . The first difference is in the construction of a min-cut cactus data structure. At the time of the Benczúr-Karger result, the fastest cactus algorithm was based on recursive contraction [1]} and had a running time of \(\tilde{O}(n^2)\) . But, this has since been improved to \(\tilde{O}(m)\)  by Karger and Panigrahi [2]}. Using this faster algorithm removes the first \(\tilde{O}(n^2)\)  bottleneck in the augmentation algorithm.
